Unconditional Probabilities

This is richer than just {p(A) | p( ) ∈ Pu< >}, allowing unions to be more precise.

Operators

We can define:

Pu<A> op Pu<B> ≡ {p → p(A) op p(B) | p( ) ∈ Pu< >},

defined where

p(A) op p(B) ∈[0,1],

so that Pu<A> op Pumaps onto possible probability values. For example, as usual, some care needs to be taken with Pu<A>/Pu, restricting the range of ‘/’. But we shall only use it where it is used in the precise theory.

Interpolation

It is not in general true that if B’ ⇒ B then P<A|B’> = P<A|B>, but it will often be the case for ‘reasonable’ B’. If we have a context C that defines a most refined partition of B, then it will often be the case that for any B’ that is a union of members of the partition, P<A|B’> = P<A|B>. For example, it may be that P(Outcome|Treatment) =p applies irrespective of gender and race but not age. This type of situation can be denoted by:

P<A|∂B:C> = … ,

meaning that P<A|B’> = … for all B’ that are parts of B with respect to a partition as determined by C.

Muddling

It is not assumed that limits such as those in the law of large number exist. If we have an indefinite sequence of epochs (such as years) that provide contexts Ci for which the P<A:Ci> are sometimes close to p and sometimes close to q ( > p) then P, where C covers all epochs, can have no limit, and is said to be muddled of degree (q-p). More generally, we can define ‘the degree of muddling’:

The usual assumption is that P⌈ :C⌉ = 0, which we can deny by stating P⌈ :C⌉ > 0 without needing to state any limits to P<A:C>. Alternatively, we often seek refined contexts C’ and specific issues, A, such that P<A:C’> is not very muddled: P⌈A:C’⌉ ≈ 0.

Often, log probability is more convenient than just probability. Here, suppose that B is a set of n equi-probabilities, independent of A. Then

P⌈A.B:C⌉ = P⌈A:C⌉/n,

so the degree of muddling depends on the granularity of the propositions, which is often arbitrary. Moreover, there is no convenient general formula. We can generalise the definition to:

In particular, if B is precise (i.e. (log.P)⌈B:C⌉=P⌈A.B:C⌉=0) then the degree of muddling of A.B is just that of A. This make (log.P)⌈_:C⌉ a much more natural measure than P⌈A.B:C⌉. I conjecture that many real-world issues are significantly muddled.